Chandan Mandal 11.April-2016. Mathematical Symbols List of all mathematical symbols and signs - meaning and examples. Basic math symbols Geometry symbols Algebra symbols Probability & statistics symbols Set theory symbols Logic symbols Calculus & analysis symbols Number symbols Greek symbols Roman numerals Basic math symbols Symbol Symbol Name Meaning / definition Example = equals sign equality 5 = 2+3 ≠ not equal sign inequality 5 ≠ 4 > strict inequality greater than 5 > 4 < strict inequality less than 4 < 5 ≥ inequality greater than or equal to 5 ≥ 4 ≤ inequality less than or equal to 4 ≤ 5 ( ) parentheses calculate expression inside first 2 × (3+5) = 16 [ ] brackets calculate expression inside first [(1+2)*(1+5)] = 18 + plus sign addition 1 + 1 = 2 − minus sign subtraction 2 − 1 = 1 ± plus - minus both plus and minus operations 3 ± 5 = 8 and -2 ∓ minus - plus both minus and plus operations 3 ∓ 5 = -2 and 8 * asterisk multiplication 2 * 3 = 6 × times sign multiplication 2 × 3 = 6 ∙ multiplication dot multiplication 2 ∙ 3 = 6 ÷ division sign / obelus division 6 ÷ 2 = 3 / division slash division 6 / 2 = 3 – horizontal line division / fraction mod modulo remainder calculation 7 mod 2 = 1 . period decimal point, decimal separator 2.56 = 2+56/100 ab power exponent 23 = 8 a^b caret exponent 2 ^ 3 = 8 √a square root √a · √a = a √9 = ±3 3√a cube root 3√a · 3√a · 3√a = a 3√8 = 2 4√a fourth root 4√a · 4√a · 4√a · 4√a = a 4√16 = ±2 n√a n-th root (radical) for n=3, n√8 = 2 % percent 1% = 1/100 10% × 30 = 3 ‰ per-mille 1‰ = 1/1000 = 0.1% 10‰ × 30 = 0.3 ppm per-million 1ppm = 1/1000000 10ppm × 30 = 0.0003 ppb per-billion 1ppb = 1/1000000000 10ppb × 30 = 3×10-7 ppt per-trillion 1ppt = 10-12 10ppt × 30 = 3×10-10 Geometry symbols Symbol Symbol Name Meaning / definition Example ∠ angle formed by two rays ∠ABC = 30º measured angle ABC = 30º spherical angle AOB = 30º ∟ right angle = 90º α = 90º º degree 1 turn = 360º α = 60º ´ arcminute 1º = 60´ α = 60º59' ´´ arcsecond 1´ = 60´´ α = 60º59'59'' line infinite line AB line segment line from point A to point B ray line that start from point A arc arc from point A to point B = 60º | perpendicular perpendicular lines (90º angle) AC |BC || parallel parallel lines AB || CD ≅ congruent to equivalence of geometric shapes and size ∆ABC ≅ ∆XYZ ~ similarity same shapes, not same size ∆ABC ~ ∆XYZ Γ triangle triangle shape ΓABC ≅ ΓBCD |x-y| distance distance between points x and y | x-y | = 5 π = 3.141592654... π pi constant c = π·d = 2·π·r is the ratio between the circumference and diameter of a circle rad radians radians angle unit 360º = 2π rad grad grads grads angle unit 360º = 400 grad Algebra symbols Symbol Symbol Name Meaning / definition Example x x variable unknown value to find when 2x = 4, then x = 2 ≡ equivalence identical to ≜ equal by definition equal by definition := equal by definition equal by definition ~ approximately equal weak approximation 11 ~ 10 ≈ approximately equal approximation sin(0.01) ≈ 0.01 ∝ proportional to proportional to y ∝ x when y = kx, k constant ∞ lemniscate infinity symbol ≪ much less than much less than 1 ≪ 1000000 ≫ much greater than much greater than 1000000 ≫ 1 calculate expression inside ( ) parentheses 2 * (3+5) = 16 first calculate expression inside [ ] brackets [(1+2)*(1+5)] = 18 first { } braces set rounds number to lower floor brackets ⌊x⌋ integer ⌊4.3⌋4 rounds number to upper ceiling brackets ⌈x⌉ integer ⌈4.3⌉5 x! exclamation mark factorial 4! = 1*2*3*4 = 24 | x | single vertical bar absolute value | -5 | = 5 f (x) function of x maps values of x to f(x) f (x) = 3x+5 f (x)=3x, g(x)=x-1 ⇒(f ∘g)(x)=3(x- (f ∘g) function composition (f ∘g) (x) = f (g(x)) 1) (a,b) open interval (a,b) = {x | a < x < b} x ∈ (2,6) [a,b] closed interval [a,b] = {x | a ≤ x ≤ b} x ∈ [2,6] ∆ delta change / difference ∆t = t1 - t0 ∆ discriminant Γ = b2 - 4ac summation - sum of all ∑ sigma ∑ x = x +x +...+x values in range of series i 1 2 n ∑∑ sigma double summation product - product of all ∏ capital pi ∏ x =x ∙x ∙...∙x values in range of series i 1 2 n e constant / Euler's e e = 2.718281828... e = lim (1+1/x)x , x→∞ number Euler- γ γ = 0.527721566... Mascheroni constant θ golden ratio golden ratio constant π = 3.141592654... π pi constant is the ratio between the c = π·d = 2·π·r circumference and diameter of a circle Linear Algebra Symbols Symbol Symbol Name Meaning / definition Example ∙ dot scalar product a ∙ b × cross vector product a × b A⊗B tensor product tensor product of A and B A ⊗ B inner product [ ] brackets matrix of numbers ( ) parentheses matrix of numbers | A | determinant determinant of matrix A det(A) determinant determinant of matrix A || x || double vertical bars norm T T A transpose matrix transpose (A )ij = (A)ji A † Hermitian matrix matrix conjugate transpose (A†)ij = (A)ji * A Hermitian matrix matrix conjugate transpose (A*)ij = (A)ji A -1 inverse matrix A A-1 = I rank(A) matrix rank rank of matrix A rank(A) = 3 dim(U) dimension dimension of matrix A rank(U) = 3 Probability and statistics symbols Symbol Symbol Name Meaning / definition Example P(A) probability function probability of event A P(A) = 0.5 probability of events probability that of events A P(A ∩ B) P(A∩B) = 0.5 intersection and B probability of events probability that of events A or P(A ∪ B) union B P(A∪B) = 0.5 conditional probability of event A given P(A | B) P(A | B) = 0.3 probability function event B occured probability density f (x) P(a ≤ x ≤ b) = ∫ f (x) dx function (pdf) cumulative F(x) distribution function F(x) = P(X ≤ x) (cdf) μ population mean mean of population values μ = 10 expected value of random E(X) expectation value E(X) = 10 variable X conditional expected value of random E(X | Y) E(X | Y=2) = 5 expectation variable X given Y variance of random variable var(X) variance var(X) = 4 X 2 2 ζ variance variance of population values ζ = 4 standard deviation of random std(X) standard deviation std(X) = 2 variable X standard deviation value of ζ standard deviation ζ = 2 X random variable X X middle value of random median variable x covariance of random cov(X,Y) covariance cov(X,Y) = 4 variables X and Y correlation of random corr(X,Y) correlation corr(X,Y) = 0.6 variables X and Y correlation of random ρ correlation ρ = 0.6 X,Y variables X and Y X,Y summation - sum of all values ∑ summation in range of series ∑∑ double summation double summation Mo mode value that occurs most frequently in population MR mid-range MR = (xmax+xmin)/2 half the population is below Md sample median this value 25% of population are below Q lower / first quartile 1 this value 50% of population are below median / second Q this value = median of 2 quartile samples 75% of population are below Q upper / third quartile 3 this value x sample mean average / arithmetic mean x = (2+5+9) / 3 = 5.333 population samples variance s 2 sample variance s 2 = 4 estimator sample standard population samples standard s s = 2 deviation deviation estimator zx standard score zx = (x-x) / sx distribution of random X ~ distribution of X X ~ N(0,3) variable X 2 N(μ,ζ ) normal distribution gaussian distribution X ~ N(0,3) U(a,b) uniform distribution equal probability in range a,b X ~ U(0,3) exponential exp(ι) f (x) = λe-λx , x≥0 distribution gamma(c, gamma distribution f (x) = λ c xc-1e-λx / Γ(c), x≥0 ι) chi-square χ 2(k) f (x) = xk/2-1e-x/2 / ( 2k/2 Γ(k/2) ) distribution F (k1, k2) F distribution k n-k Bin(n,p) binomial distribution f (k) = nCk p (1-p) Poisson(ι) Poisson distribution f (k) = λke-λ / k! geometric Geom(p) f (k) = p (1-p) k distribution hyper-geometric HG(N,K,n) distribution Bernoulli Bern(p) distribution Combinatorics Symbols Symbol Symbol Name Meaning / definition Example n! factorial n! = 1·2·3·...·n 5! = 1·2·3·4·5 = 120 nPk permutation 5P3 = 5! / (5-3)! = 60 nCk combination 5C3 = 5!/[3!(5-3)!]=10 Set theory symbols Symbol Symbol Name Meaning / definition Example A = {3,7,9,14}, { } set a collection of elements B = {9,14,28} objects that belong to set A and intersection A ∩ B set B A ∩ B = {9,14} objects that belong to set A or A ∪ B = A ∪ B union set B {3,7,9,14,28} subset has fewer elements or {9,14,28} ⊆ A ⊆ B subset equal to the set {9,14,28} subset has fewer elements than {9,14} ⊂ A ⊂ B proper subset / strict subset the set {9,14,28} {9,66} ⊄ A ⊄ B not subset left set not a subset of right set {9,14,28} set A has more elements or {9,14,28} ⊇ A ⊇ B superset equal to the set B {9,14,28} proper superset / strict set A has more elements than {9,14,28} ⊃ A ⊃ B superset set B {9,14} {9,14,28} ⊅ A ⊅ B not superset set A is not a superset of set B {9,66} 2A power set all subsets of A power set all subsets of A A={3,9,14}, both sets have the same A = B equality B={3,9,14}, members A=B all the objects that do not Ac complement belong to set A objects that belong to A and not A = {3,9,14}, A \ B relative complement to B B = {1,2,3}, A-B = {9,14} A = {3,9,14}, objects that belong to A and not A - B relative complement B = {1,2,3}, to B A-B = {9,14} A = {3,9,14}, objects that belong to A or B B = {1,2,3}, A ∆ B symmetric difference but not to their intersection A ∆ B = {1,2,9,14} A = {3,9,14}, objects that belong to A or B B = {1,2,3}, A ⊖ B symmetric difference but not to their intersection A ⊖ B = {1,2,9,14} A={3,9,14}, 3 ∈ a∈A element of set membership A A={3,9,14}, 1 ∉ x∉A not element of no set membership A (a,b) ordered pair collection of 2 elements set of all ordered pairs from A A×B cartesian product and B A={3,9,14}, |A| cardinality the number of elements of set A |A|=3 A={3,9,14}, #A cardinality the number of